homotopy
Sign in to savethumb|The two dashed Path (topology)|paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
Wikidata facts
- Subclass of
- continuous function
- Image
- Homotopy between two paths.svg
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- studied by
- homotopy theory
- Commons category
- Homotopy
- on focus list of Wikimedia project
- Wikipedia:Vital articles/Level/4
- maintained by WikiProject
- WikiProject Mathematics
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~17 min read
Encyclopedic overview
20 sectionsContents
- Formal definition
- Properties
- Examples
- Homotopy equivalence
- Homotopy equivalence vs. homeomorphism
- Examples
- Null-homotopy
- Invariance
- Variants
- Relative homotopy
- Isotopy
- Timelike homotopy
- Properties
- Lifting and extension properties
- Groups
- Homotopy category
- Applications
- See also
- References
- Sources
thumb|The two dashed Path (topology)|paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
Excerpted from Wikipedia’s “homotopy” article, available under the CC BY-SA 4.0 licence.